Label
Class
Class size
Class degree
Base field
Field degree
Field signature
Conductor
Conductor norm
Discriminant norm
Root analytic conductor
Bad primes
Rank
Torsion
CM
CM
Sato-Tate
$\Q$-curve
Base change
Semistable
Potentially good
Nonmax $\ell$
mod-$\ell$ images
$Ш_{\textrm{an}}$
Tamagawa
Regulator
Period
Leading coeff
j-invariant
Weierstrass coefficients
Weierstrass equation
21.1-a2
21.1-a
$6$
$8$
4.4.9909.1
$4$
$[4, 0]$
21.1
\( 3 \cdot 7 \)
\( 3^{2} \cdot 7^{16} \)
$13.01460$
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2^{2} \)
$4.256337325$
$87.10503032$
3.724468981
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \)
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16600 a^{2} + 62447 a - 34158\) , \( 1313070 a^{3} - 1585569 a^{2} - 5963840 a + 3262240\bigr] \)
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16600a^{2}+62447a-34158\right){x}+1313070a^{3}-1585569a^{2}-5963840a+3262240$
21.1-d4
21.1-d
$6$
$8$
4.4.9909.1
$4$
$[4, 0]$
21.1
\( 3 \cdot 7 \)
\( 3^{2} \cdot 7^{16} \)
$13.01460$
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$4$
\( 2^{2} \)
$3.309118906$
$22.08448528$
2.936599577
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \)
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -34 a^{3} + 6 a^{2} + 144 a - 69\) , \( -240 a^{3} + 410 a^{2} + 1244 a - 706\bigr] \)
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-34a^{3}+6a^{2}+144a-69\right){x}-240a^{3}+410a^{2}+1244a-706$
Download
displayed columns for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.